Determinant and Inverse Matrix - NYU Courant

[Pages:5]Determinant and Inverse Matrix

Liming Pang

Definition 1. A n ? n square matrix A is invertible if there exists a n ? n matrix A-1 such that AA-1 = A-1A = In, where In is the identity n ? n matrix. If A-1 exists, we say A-1 is the inverse matrix of A.

Proposition 2. If A and B are n ? n matrices, then AB = In BA = In.

Example 3.

83 52

2 -5

-3 8

=

1 0

0 1

=

2 -5

-3 8

83 52

So

8 5

3 2

is invertible and its inverse is

2 -5

-3 8

Remark 4. If A is invertible, then it follows directly from definition that A-1 is also invertible and the inverse of A-1 is A.

Proposition 5. If A, B are n ? n matrices, then:

1. (A-1)-1 = A

2. (AB)-1 = B-1A-1

3. (AT )-1 = (A-1)T

It is a natural question to ask if there is some way to tell if a matrix is invertible before looking for its inverse. It turns out that the concept of determinant solve the problem completely.

We will define determinant of a n ? n matrix in a recursive manner.

Definition 6. A is a n ? n square matrix, where n > 1. Define the matrix Aij to be the (n - 1) ? (n - 1) square matrix obtain from A by deleting the i-th row and j-th column.

1

 2 -3 1

Example 7. Let A = -5 8 9 ,

4 7 -2

-5 9

2 -3

then A12 = 4 -2 , A33 = -5 8

Definition 8. If A = {a} is a 1 ? 1 matrix, define the determinant of A to be the number

det(A) = a

If A = {aij} is a n ? n (n > 1) square matrix, then define the determinant of A to be the number

n

det(A) = (-1)i+1ai1 det Ai1

i=1

We now take the case n = 2: If A = {aij} is a 2 ? 2 matrix, then bu definition,

2

det(A) = (-1)i+1ai1 det(Ai1) = a11 det A11 - a21 det A21 = a11a22 - a21a12

i=1

So we have obtained the formula

det

a c

b d

= ad - bc

(0.1)

We can thus directly apply Formula 0.1 when computing the determinant of a 2 ? 2 matrix.

Example 9.

det

1 4

-5 2

= 1 ? 2 - 4 ? (-5) = 22

Next let's compute the determinant of a 3 ? 3 matrix:

Example 10.

1 det 4

7

2 5 8

3 6 = 1 ? det 9

5 8

6 9

- 4 ? det

2 8

3 9

+ 7 ? det

2 5

3 6

= 1 ? (5 ? 9 - 6 ? 8) - 4 ? (2 ? 9 - 3 ? 8) + 7 ? (2 ? 6 - 3 ? 5) = 1 ? (-3) - 4 ? (-6) + 7 ? (-3) =0

2

There are some interesting properties regarding determinants: Proposition 11. Let A be an n ? n matrix, then:

1. det(In) = 1 2. det(A) = det(AT ).

3. If all entries in some row (column) are 0, then det(A) = 0.

4. If B is obtained from A by switching two rows (or columns), then det(B) = - det(A).

5. If B is obtained from A by multiplying some number to all the entries in one of the rows (columns), then det(B) = det(A).

6. If B is obtained from A by adding a multiple of one row (column) to another, then det(A) = det(B).

7. If there are two rows (columns) of A such that one is a multiple of another, then det(A) = 0

8. det is multiplicative: for any n ? n matrices A and B, det(AB) = det(A) det(B)

1 1 3

Example 12. By the above propositions, we see that det 4 4 6 = 0.

779

1 4 3

Then we add 1 times the third column to the second column, det 4 10 6 =

7 16 9

1 2 3

0.

Next

we

multiply

the

second

column

by

1 2

,

we

get

det 4

5

6

=

1 2

?0

=

789

0, which agrees with the previous example.

Remark 13. The above propositions implies we can compute determinant in a more general way:

n

n

det(A) = (-1)i+jaij det(Aij) = (-1)i+j det(Aij)

i=1

j=1

3

An important application of determinant is that it can be used to test if a square matrix is invertible or not:

Theorem 14. A square matrix A is invertible if and only if det(A) = 0.

1 2 3

1 2 3

Example 15. We have shown that det 4 5 6 = 0, so 4 5 6 is not

789

789

invertible.

Example 16. A 2 ? 2 matrix

a c

b d

is invertible if and only if

det(

a c

b d

) = ad - bc = 0

The method of Gaussian Elimination can be applied to computing the inverse of a given matrix.

Theorem 17. A is an invertible n ? n matrix. If we form the matrix

A In

a11 ... a1n 1 0 ... 0

=

a21 ... a2n 0 1 ... 0 .........................

an1 ... ann 0 0 ... 1

and apply the method of Gaussian Elimination, we will obtain

In A-1

Example 18. Find the inverse of

1 3 3 1 3 4

143

1 3 3 1 0 0 1 3 3 1 0 0 1 3 3 1 0 0

1 3 4 0 1 0 0 0 1 -1 1 0 0 1 0 -1 0 1

143001

0 1 0 -1 0 1

0 0 1 -1 1 0

1 0 3 4 0 -3 1 0 0 7 -3 -3

0 1 0 -1 0 1 0 1 0 -1 0 1

0 0 1 -1 1 0

0 0 1 -1 1 0

4

So we find the inverse of the given matrix is

7 -3 -3 -1 0 1

-1 1 0

When an invertible matrix is 2 ? 2, there is a formula for its inverse:

Proposition 19. If

a c

b d

is an invertible matrix, then its inverse is

1 d -b ad - bc -c a

Example 20. Find the inverse of

1 3

2 4

1 2 -1

1

4 -2 -2 1

34

= 1?4-2?3

-3

1

=

3 2

-

1 2

Proposition 21. For a n ? n diagonal matrix A = (aij) (i.e. aij = 0 for any i = j), its determinant is

Example 22.

det(A) = a11a22...ann

2 0 0 det 0 3 0 = 2 ? 3 ? (-4) = -24

0 0 -4

5

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